The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a compact Hausdorff space, then $RR$ $C(X)=dim(X)$.

Let $A$ be a $C^*$-algebra. we define $RR(A)=0$ iff every self-adjoint element in $A$ can be approximate by invertible, self-adjoint element in $A$. for example every Von-Neumman algebra has real rank zero.

So, on the other hand since $C^*(\mathbb T)=C(\mathbb Z)$ and $C^*(\mathbb Z)=C(\mathbb T)$, then $RR$ $C^*(\mathbb T)=0$ but

$RR$ $C^*(\mathbb Z)$$\not=$$0$ .

**Question:** Which locally compact groups $G$ satisfy $RR$ $C^*(G)=0$?

*NOTE:* If $G$ is locally compact group with fixed Harr measure, let full group $C^*$-algebra $C^*(G)$ be the completion of the convolution algebra $L^1(G)$ with respect to the norm $\lVert f\rVert_{C^*(G)}$=$\sup $$\{\lVert\phi(f)\rVert\}$, where the supremum is taken over all $*$-representation $\phi$ of $L^1(G)$ as an algebra of bounded linear operators in a Hilbert space.